We propose a non-linear, hybrid quantum-classical scheme for simulating non-equilibrium dynamics of strongly correlated fermions described by the Hubbard model in a Bethe lattice in the thermodynamic limit. Our scheme implements non-equilibrium dynamical mean field theory (DMFT) and uses a digital quantum simulator to solve a quantum impurity problem whose parameters are iterated to self-consistency via a classically computed feedback loop where quantum gate errors can be partly accounted for. We analyse the performance of the scheme in an example case.

f1: (a) In non-equilibrium DMFT a fermionic quantum lattice model is replaced by a single impurity site exchanging particles via a self-consistently determined time and spin dependent mean field Λσ(t, t′). (b) This exchange of particles yields dynamical fluctuations of the impurity site occupation as a function of time shown here as /↑〉 → /↓↑〉 → /↓〉 →/vac〉. The onsite interaction U energetically penalises the doubly occupied state /↓↑〉. (c) The impurity-mean field interaction is mapped onto a SIAM with unitary evolution . The energies of the non-interacting bath sites p are chosen for t > 0 and their chemical potential is set μ = 0 in this work24. The impurity site exchanges fermions with time-dependent hybridization energies Vpσ(t). (d) Quantum-classical hybrid simulation scheme: the SIAM dynamics for a given set of parameters Vpσ(t) is implemented on a quantum coprocessor and yields the impurity Green’s function Gσ(t, t′). The classical non-linear feedback loop takes Gσ(t, t′) and calculates the mean field Λσ(t, t′) from which a new set of Vpσ(t) can be extracted. These parameters are then fed back into the quantum coprocessor and the loop is repeated until self-consistency is achieved.

Mentions:
This and similar models are extremely challenging to study numerically due to the exponential growth of the Hilbert space with system size. One thus often resorts to mean field approximations which typically consider only a single lattice site and replace interactions with its neighbourhood by a mean field Λ. This turns a linear quantum problem in an exponentially large Hilbert space into a much smaller but non-linear problem where Λ needs to be determined self-consistently. Such mean field approximations become increasingly accurate with the number of nearest neighbours. A classic example of this approach is the Weiss theory of ferromagnetism21. For mean field theory to be applicable to strongly correlated Fermi systems in thermal equilibrium, the mean field Λσ(t) has to be dynamical to account for correlations between interactions with the environment that are separated by t in time, as schematically shown in Fig. 1a,b.

f1: (a) In non-equilibrium DMFT a fermionic quantum lattice model is replaced by a single impurity site exchanging particles via a self-consistently determined time and spin dependent mean field Λσ(t, t′). (b) This exchange of particles yields dynamical fluctuations of the impurity site occupation as a function of time shown here as /↑〉 → /↓↑〉 → /↓〉 →/vac〉. The onsite interaction U energetically penalises the doubly occupied state /↓↑〉. (c) The impurity-mean field interaction is mapped onto a SIAM with unitary evolution . The energies of the non-interacting bath sites p are chosen for t > 0 and their chemical potential is set μ = 0 in this work24. The impurity site exchanges fermions with time-dependent hybridization energies Vpσ(t). (d) Quantum-classical hybrid simulation scheme: the SIAM dynamics for a given set of parameters Vpσ(t) is implemented on a quantum coprocessor and yields the impurity Green’s function Gσ(t, t′). The classical non-linear feedback loop takes Gσ(t, t′) and calculates the mean field Λσ(t, t′) from which a new set of Vpσ(t) can be extracted. These parameters are then fed back into the quantum coprocessor and the loop is repeated until self-consistency is achieved.

Mentions:
This and similar models are extremely challenging to study numerically due to the exponential growth of the Hilbert space with system size. One thus often resorts to mean field approximations which typically consider only a single lattice site and replace interactions with its neighbourhood by a mean field Λ. This turns a linear quantum problem in an exponentially large Hilbert space into a much smaller but non-linear problem where Λ needs to be determined self-consistently. Such mean field approximations become increasingly accurate with the number of nearest neighbours. A classic example of this approach is the Weiss theory of ferromagnetism21. For mean field theory to be applicable to strongly correlated Fermi systems in thermal equilibrium, the mean field Λσ(t) has to be dynamical to account for correlations between interactions with the environment that are separated by t in time, as schematically shown in Fig. 1a,b.

We propose a non-linear, hybrid quantum-classical scheme for simulating non-equilibrium dynamics of strongly correlated fermions described by the Hubbard model in a Bethe lattice in the thermodynamic limit. Our scheme implements non-equilibrium dynamical mean field theory (DMFT) and uses a digital quantum simulator to solve a quantum impurity problem whose parameters are iterated to self-consistency via a classically computed feedback loop where quantum gate errors can be partly accounted for. We analyse the performance of the scheme in an example case.